Not Updated For Current Season
First of all I just want to preface with saying that this is my first post ever. I am not that experienced with League of Legends so please leave comments and suggestions.
This build is to take advantage of two aspects of Jax's character and that is survivability and hitting really hard. This is by no means a universal build. As with any character in LOL you have to build to your opponent. I will discuss some other options that I have tried and likes as well that are situational. I will not be going into his abilities with much depth so I suggest reading the tooltips :)
I like to go all the way in the offensive tree making sure to get spell pen for his ult and the little AP that you will pick up along the way. The rest is pretty much self explanatory. Also, since Jax is a dodge based character I like to get up to nimbleness in the defensive tree. This allows you a slight advantage in combat when you block an attack and they try to get away once you put your boot up their a**.
As for runes I like taking dodge quints and seals as well as attack speed glyphs and marks. these will help you early game to have a better chance at getting first blood or just being able to harass and stay in the lane, depending on your play style. I have also seen people take armor penetration which never hurts since he is basically an auto attacker besides his highly damaging moves (which include physical damage). However, I have found that the attack speed is helpful since in this build you don't get the bulk of it until later in the game.
My item choices are pretty mixed for a Jax build. Because of the fact that he gets his bonus damage from both ability power and attack damage he is a prime candidate for some of these items. I will go through them all with you now.
I like to start off with boots and three health potions. This gives me an advantage when chasing enemies to get first blood. It also allows you do harass a little more because you can more easily duck in and out of combat with his leap strike ability.
I wait until I have enough gold to get my ninja tabi boots and maybe a long sword that will later be built into a phage. I go with the phage off the bat because it helps tremendously when chasing and if you want to empower spam you need to be right on their butts...because they WILL be running away from you :)
Like I said I then get a phage, zeal (for attack speed and crit), then a sheen which I will turn into a Trinity Force. The tri-force is a great item for Jax, allowing him to use his abilities and then hit for 150% of his attack power on his next auto attack. Obviously it has other benefits as well. Tri-force is a must have in any Jax build (IMO).
Next I like to get the Hextech Gunblade. I start by building the attack damage side of the build. You could start with the spell vamp side but I think at this point a vampiric scepter is a life saver. You will need this to harass, farm creeps and stay in the lane until you can get the more expensive items. Plus the bilgewater cutlass has a use ability which provides a little more damage output but more importantly a slowing ability. I'm not sure if it stacks with tri-force but when they are just out of range and your leap is on cooldown you can bet that this will come in handy.
After the Hextech I like to get a Guinsoo's Rageblade. The rageblade along with your ult will make you attack extremely fast and the unique attack speed and ability power bonus of the rageblade will make you much more powerful in one on one combat and even in team fights when you don't get focused.
I would consider all of the above to be Jax's core items. With this combination you are one of the best one vs. one characters in the game. However, the next items are more specific to one type of build, and that is the tank destroyer build. Since Jax is so powerful and tough to kill I like the idea of being able to nuke down a tank when you get a chance. This will put your team in a great position when the opposing team is without a tank for a minute at level 18. To do this however you need need more attack speed to get your bonuses from your ult and your rageblade faster. You need critical strike so you hit him harder and the best thing you can get is something that does percentages of damage. For all of these qualities I like the phantom dancer and the madred's bloodrazor. These items combined will give you a huge increase in damage output, especially on tanks, support classes with a lot of health or shields like Karma or even just a vladimir. The madred's gives you an extra 4% of their health in magic damage and the crit and attack speed from the phantom dancer can't be topped. Not to mention you will be moving even faster with the movement speed increase from the dancer as well. It's a great choice for this type of Jax.
Obviously this is a situational thing. If for instance you have a tank killer on your team who is doing well then you should think about getting something else. I would suggest phantom dancer infinity edge. You will have an outrageous amount of attack speed and the infinity edge will make you hit extremely hard.
That's all for me. I don't really want to get into the actually play style for Jax. You can figure that out on your own or read a different guide. :) Comments and critiques are welcome. I want to improve as well. Thanks for reading!